Claudia Castro-Castro
Math 283 Spring 2020
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DEFINITION Let \( D \) be a set in \( \mathbb{R}^2 \) (a plane region). A vector field on \( \mathbb{R}^2 \) is a function \( \mathbf{\vec{F}} \) that assigns to each point \( (x,y) \) in \( D \) a two-dimensional vector \( \mathbf{\vec{F}}(x,y) \)
Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition
| \( (x,y) \) | \( \mathbf{\vec{F}}(x,y) \) |
|---|---|
| \( (0,0) \) | \( \langle 1,\;-1\rangle \) |
| \( (x,y) \) | \( \mathbf{\vec{F}}(x,y) \) |
|---|---|
| \( (0,0) \) | \( \langle 1,\;-1\rangle \) |
| \( (1,0) \) | \( \langle 1,\;-1\rangle \) |
| \( (2,0) \) | \( \langle 1,\;-1\rangle \) |
| \( (0,1) \) | \( \langle 1,\;-1\rangle \) |
| \( (1,1) \) | \( \langle 1,\;-1\rangle \) |
| \( (2,1) \) | \( \langle 1,\;-1\rangle \) |
| \( \vdots \) | \( \vdots \) |
| \( (x,y) \) | \( \mathbf{\vec{F}}(x,y)=\langle -y,x\rangle \) |
|---|---|
| \( (x,y) \) | \( \mathbf{\vec{F}}(x,y)=\langle -y,x\rangle \) |
|---|---|
| \( (0,0) \) | \( \langle 0,\;0\rangle \) |
| \( (1,0) \) | \( \langle -0,\;1\rangle \) |
| \( (1,1) \) | \( \langle -1,\;1\rangle \) |
| \( (0,1) \) | \( \langle -1,\;0\rangle \) |
| \( (0,2) \) | \( \langle -2,\;0\rangle \) |
| \( (-1,1) \) | \( \langle -1,\;-1\rangle \) |
| \( (-1,0) \) | \( \langle 0,\;-1\rangle \) |
| \( (-1,-1) \) | \( \langle 1,\;-1\rangle \) |
| \( (0,-1) \) | \( \langle 1,\;0\rangle \) |
| \( (1,-1) \) | \( \langle 1,\;1\rangle \) |
\[ \mathbf{\hat{F}}(x,y)=\langle x,\;y\rangle \]
\[ \mathbf{\hat{F}}(x,y)=\langle y,\;\sin{x}\rangle \]
\[ \mathbf{\hat{F}}(x,y)=\langle x^2-y^2,\;2xy\rangle \]
\[ \mathbf{\hat{F}}(x,y)=mMG\frac{\langle -x,-y\rangle}{(x^2+y^2)^{\frac{3}{2}}} \]
Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition
DEFINITION Let \( E \) be a set in \( \mathbb{R}^3 \). A vector field on \( \mathbb{R}^3 \) is a function \( \mathbf{\vec{F}} \) that assigns to each point \( (x,y,z) \) in \( E \) a three-dimensional vector \( \mathbf{\vec{F}}(x,y,z) \)
Courtesy of James Stewart, Calculus: Early transcendentals, 2nd edition
\[ \mathbf{\hat{F}}(x,y,z)=\langle 1,\;1,-1\rangle \]
\[ \mathbf{\hat{F}}(x,y,z)=mMG\frac{\langle -x,\;-y,-z\rangle}{(x^2+y^2+z^2)^{\frac{3}{2}}} \]
Courtesy of J. Stewart, Calculus: Early transcendentals, 2nd edition